Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $n \neq 0$. $r = \dfrac{n^2 - 6n + 5}{-7n - 42} \div \dfrac{n - 1}{n + 6} $
Dividing by an expression is the same as multiplying by its inverse. $r = \dfrac{n^2 - 6n + 5}{-7n - 42} \times \dfrac{n + 6}{n - 1} $ First factor the quadratic. $r = \dfrac{(n - 1)(n - 5)}{-7n - 42} \times \dfrac{n + 6}{n - 1} $ Then factor out any other terms. $r = \dfrac{(n - 1)(n - 5)}{-7(n + 6)} \times \dfrac{n + 6}{n - 1} $ Then multiply the two numerators and multiply the two denominators. $r = \dfrac{ (n - 1)(n - 5) \times (n + 6) } { -7(n + 6) \times (n - 1) } $ $r = \dfrac{ (n - 1)(n - 5)(n + 6)}{ -7(n + 6)(n - 1)} $ Notice that $(n + 6)$ and $(n - 1)$ appear in both the numerator and denominator so we can cancel them. $r = \dfrac{ \cancel{(n - 1)}(n - 5)(n + 6)}{ -7(n + 6)\cancel{(n - 1)}} $ We are dividing by $n - 1$ , so $n - 1 \neq 0$ Therefore, $n \neq 1$ $r = \dfrac{ \cancel{(n - 1)}(n - 5)\cancel{(n + 6)}}{ -7\cancel{(n + 6)}\cancel{(n - 1)}} $ We are dividing by $n + 6$ , so $n + 6 \neq 0$ Therefore, $n \neq -6$ $r = \dfrac{n - 5}{-7} $ $r = \dfrac{-(n - 5)}{7} ; \space n \neq 1 ; \space n \neq -6 $